MATHEMATICS HIGH SCHOOL

The planets in our solar system do not travel in circular paths. Rather, their orbits are elliptical. The Sun is located at a focus of the ellipse. The perihelion is the point in a planet’s orbit that is closest to the Sun. So, it is the endpoint of the major axis that is closest to the Sun.

The aphelion is the point in the planet’s orbit that is furthest from the Sun. So, it is the endpoint of the major axis that is furthest from the Sun.

The closest Mercury comes to the Sun is about 46 million miles. The farthest Mercury travels from the Sun is about 70 million miles.

1. What is the distance between the perihelion and the aphelion?

2. What is the distance from the center of Mercury’s elliptical orbit and the Sun?

3. Write the equation of the elliptical orbit of Mercury, where the major axis runs horizontally. Allow a and b to be measured in millions of miles. Use the origin as the center of the

Answers

Answer 1

Answer:

Answer:

(1) 83.764 million miles

(2) 52.766 million miles

(3) $(x^2)/(4900)+(y^2)/(2116)=1.$

Step-by-step explanation:

Let the origin C(0,0) be the center of the elliptical path as shown in the figure, where the location of the sun is at one of the two foci, say f.

The standard equation of the ellipse having the center at the origin is

$(x^2)/(a^2)+(y^2)/(b^2)=1\;\cdots(i)$

where $a$ and $b$ are the semi-axes of the ellipse along the x-axis and y-axis respectively.

Let the points P and A represent the points of perihelion (nearest to the sun) and the aphelion (farthest to the sun) of the closest planet Mercury.

Given that,

CP=46 million miles and

CA=70 million miles.

So, $CP=b$ is the semi-minor axis and $CA=a$ is the semi-major axis.

Let the distances on the axes are in millions of miles. So, the coordinates of the point P and A are $P(0,46)$ and $A(70,0)$ respectively.

(1) From the distance formula, the distance between the perihelion and the aphelion is

$PA=√((0-70)^2+(46-0)^2)=83.764$ million miles.

(2) Location of the Sun is at focus, $f$ , of the elliptical path.

From the standard relation, the distance of the focus from the center of the ellipse, c, is

$c=ae\;\cdots(ii)$

where $a$ and $e$ are the semi-major axis and the eccentricity of the ellipse.

The eccentricity of the ellipse is

$e=\sqrt{1-(b^2)/(a^2)}$

$\Rightarrow e=\sqrt{1-(46^2)/(70^2)}=0.7538$ .

Hence, from the equation (i) the distance of the Sun from the center of the elliptical path of the Mercury is

$c=70*0.7538=52.766$ million miles.

(3) From the equation (i), the equation of the elliptical orbit of Mercury is

$(x^2)/(70^2)+(46^2)/(b^2)=1$

$\Rightarrow (x^2)/(4900)+(y^2)/(2116)=1.$

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"The inverse of every function must be a function itself." State whether you agree or disagree with the statement providing evidence to support your position.
(1)/(x)+(1)/(2x+1) = (1)/(3.5)SOLVE WITH STEPS PLS I NEED IT QUICK
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A, B, and C are polynomials, where A = n, B = 2n + 6, and C = n2 – 1. What is AB – C in simplest form?A=–n2 + 3n + 5 B=n2 + 6n + 1 C=2n2 + 6n – 1 D=3n2 + 5

2r-4/r-2 please help

Answers

$(2r-4)/(r-2 )\n \n r-2 \neq 0 \n \nr\neq 2\n\n\n(2r-4)/(r-2 ) =(2(r-2))/((r-2) ) = 2 \n \n$

2r-4/r-2 = 2(r-2)/r-2 = 2

Line BC reflects about a line such that N is the reflection of B and O is the reflection of C. Point N is shown on the coordinate plane, but point O is not. What will the coordinates of point O be?(1, 5)
(3, 5)
(5, 5)
(6, 5)
(5,5)
(3,5)

Answers

If the line BC reflects about a line and point N( 3, 5 ) is the reflection of point B ( 3 , 7 ) then the reflection of C ( 5, 7 ) is O ( 5 , 5 ).
Answer: C ) ( 5, 5 )

Answer:

(5,5)

Step-by-step explanation:

Given that B has coordinates as (3,7) When reflected about a line the new coordinates are (3,5)

This implies that by reflection x coordinate remains the same but y reduced by 2 units.

Apply the same logic for unknown O.

C is reflected to O.

Original coordinates of C are (5,7)

Hence new coordinates i.e. that of O would be

x coordinate same as 5

y coordinate 2 less = 7-2 = 5

COordinates of O are (5,5)

A sugar bowl holds 237 grams. You have a one kilogram bag of sugar. Estimate how many bowls of sugar you can fill from the bag.

Answers

Answer:

The required number of bowls are 5.

Step-by-step explanation:

Given : A sugar bowl holds 237 grams. You have a one kilogram bag of sugar.

To find : Estimate how many bowls of sugar you can fill from the bag?

Solution :

1 bowl can hold 237 gram of sugar.

We have, 1 kg of sugar or 1000 gram of sugar.

According to question,

237 gram of sugar can hold in 1 bowl.

So, 1 gram of sugar can hold in $(1)/(237)$ bowl.

1000 gram of sugar can hold in $(1000)/(237)$ bowl.

1000 gram of sugar can hold in $4.21$ bowl.

Which means, The required number of bowls are 5.

As 4 bowls have $237* 4=948$ grams of sugar.

Sugar left is 1000-948=52 grams

That 52 grams is filled into 5th bowl.

7÷3×5x÷12. Can someone help me multiply rational expressions and divide rational expressions? Please and thank you.

Answers

In this case, because there are no exponents or brackets, (BEDMAS) you would just do it in the order given.

7÷3×5x÷12= 7×1/3 × 5x × 1/12
To divide, make the numbers after the divide sign into fractions.(ex: 3= 1/3)
Then multiply it out:
= 7/3 × 5x/12
= 35x/34
Then because the numerator is bigger then the denominator, make the improper
fraction into a mixed fraction:
= 1 1x/34

Hope this helps!

7 ÷ 3 · 5x ÷ 12
7 ÷ 15x ÷ 12
7 ÷ 5x/4
28x/5

What is 5 x 2 + 3 to the power of 2

Answers

5*2 = 10

3^2 = 3*3 = 9

10 + 9 = 19

Answer: 19

Maggie and her friends are each able to recruit a certain number of people each year to sample a product. The number of people per year is represented by the function f(t) = 3(2)t. What does the t represent? How many people were recruited in 6 years?.

Answers

If the function is:
$f ( x ) = 3 * 2^(t)$
t represents time ( in years ).
In 6 years :
$f ( 6 ) = 3*2 ^(6) = 3* 64 = 192$
Answer:
B ) t represents the number of year; 192 people were recruited in 6 years.

Answer:

B. t represents the number of years ; 192 people were recruited in 6 years

Step-by-step explanation:

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